Marcel Riesz’s Work on Clifford Algebras

Abstract

This article reviews Marcel Riesz’s lecture notes on Clifford Numbers and Spinors, 1958, and evaluates its effect on present research on Clifford algebras.

The article begins with a critical survey of the history of Clifford algebras. Inaccuracies in citations are pointed out and mistaken priorities are rectified. Misconceptions about Clifford algebras are examined and flaws are corrected. The paper focuses on controversial issues that have been debated over the past few years, and offers definitive resolutions. Concrete counter-examples are given to misplaced claims.

Particular attention is directed to the existence of a canonical linear isomorphism between the exterior algebra and the Clifford algebra. Chevalley constructed the Clifford algebra Cl(Q) of a quadratic form Q as a subalgebra of the endomorphism algebra End(⋀V) of the exterior algebra ⋀V. This construction depends on an arbitrary, not necessarily symmetric, bilinear form B such that B(x, x) = Q(x). The choice of B fixes the contraction u⌋v in ⋀V and permits the introduction of a Clifford product xu = x⌋u + x⋀u of x ∈ V and u ∈ ⋀ V. This gives rise to the Clifford algebra of the symmetric bilinear form 1/2(B(x,y)+B(y,x)) when the characteristic ≠ 2.

Marcel Riesz went backwards with his fundamental formulas

$$x\left. {\underline {\, {} \,}}\! \right| \;u\frac{1}{2}(xu - \hat ux)and\;x\Lambda u = \frac{1}{2}(xu + \hat ux)$$

and re-introduced the contraction and the exterior product in the Clifford algebra, and constructed, in all characteristics ≠ 2, a privileged linear isomorphism from the Clifford algebra to the exterior algebra (without antisymmetric products of vectors). [In the above $\hat u$ means the grade involute of u, that is, for a homogeneous element a ∈ ⋀ ^k V we have ${\hat a}$ = (-1)^ka.]